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\section*{Homework 5}\renewcommand{\leftmark}{Homework 5}\phantomsection\addcontentsline{toc}{section}{Homework 5}

\begin{exercise}
How many digits (in the standard decimal presentation) does the
number $1000!$ have? Find several (as many as you can) first
digits. 
\end{exercise}
\begin{exercise}
Prove that for any real $a$, $b$, $\|\Gamma(a +
b\I)\|\leq\|\Gamma(a)\|$. 
\end{exercise}
\begin{exercise}
Prove that
$\displaystyle\Gamma(x)=\frac{1}{x}\int^{\infty}_{0}\E^{-t^{1/x}}\D{t}$.
(This is a generalization of the equality
$\displaystyle\Gamma\left(\frac{1}{2}\right)=2\int^{\infty}_{0}\E^{-t^{2}}\D{t}$.)
\end{exercise}
\begin{exercise}
Let $\displaystyle
f(x)=\int^{\infty}_{x}\frac{\E^{x-t}}{t}\D{t}$, $\displaystyle S_{n}(x)=\frac{1}{x}-\frac{1}{x^{2}}+\dots+\frac{(-1)^{n-1}(n-1)!}{x^{n}}$.
The ``relative error'' of the approximation $f(x)\approx
S_{n}(x)$ is $\displaystyle\left|\frac{f(x)-S_{n}(x)}{f(x)}\right|$.

(a) Prove that there exists a sequence $\varepsilon_{n}$, $\lim_{n\to\infty}\varepsilon_{n}=0$,
such that if $x>n$, then the relative error of the approximation
$f(x)\approx S_{n}(x)$ does not exceed $\varepsilon_{n}$. 

(b) Is it true (or likely to be true) that the inequality $x>n$
in the previous statement can be replaced by $x>\sqrt{n}$?
\end{exercise}
